They are the same. Burnup refers to the energy produced per unit mass of fuel, usually in MWd/kgU or GWd/tU, although the Canadians like to use MWh/kgU, the Belgians and some others used to use MWd/kgUO[sub]2[/sup], and for a long time GE (GNF) used GWd/stU (st = short ton). In MOX cores, the burnup may expressed in GWd/tHM (HM=Heavy Metal, U+Pu). Finally some folks used FIMA, or fissions per initial metal atom, with a rough equivalence of 1% FIMA = 9.75 GWd/tU.
The term depletion refers to the reduction or depletion of enrichment of the fuel. When fuel is irradiated, most of the fission event occur in U-235 until sufficient Pu-239/Pu-240/Pu-241 build up to compete with the U-235 for neutrons.
Using a code like SIMULATE, one does core depletion calculations which basic simulate the fission process in the core during a cycle of operation.